Weakening of Intuitionistic Negation for Many-valued Paraconsistent da Costa System

Abstract. In this paper we propose substructural propositional logic obtained by da Costa weakening of the intuitionistic negation. We show that the positive frag-ment of the da Costa system is distributive lattice logic, and we apply a kind of da Costa weakening of negation, by preserving, differently from da Costa, its funda-mental properties: antitonicity, inversion and additivity for distributive lattices. The other stronger paraconsistent logic with constructive negation is obtained by adding an axiom for multiplicative property of weak negation. After that we define Kripke style semantics based on possible worlds, and derive from it many-valued semantics based on truth-functional valuations, for these two paraconsis-tent logics. Finally we demonstrate that this model-theoretic inference system is adequate: sound and complete w.r.t. the axiomatic da Costa like systems for these two logics.